Simultaneous estimation of two SU(2) parameters: A study of optimal states and precision bounds

Pal, Ritopriyo (2025) Simultaneous estimation of two SU(2) parameters: A study of optimal states and precision bounds. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Ritopriyo Pal (20MS070)) 20MS070_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

In quantum parameter estimation problems, the minimum possible error with which a particular quantum state can be used to estimate certain parameter(s) of interest is given by appropriate figures of merit, which define fundamental lower bounds for the error in such an estimation protocol. These quantities or lower bounds are usually defined in a way such that they incorporate optimizations over all possible measurements as well as all possible estimator functions. For example, in the case of quantum single-parameter estimation, the minimum error corresponds to the inverse of the so called quantum Fisher information, and the corresponding lower bound is known as the single-parameter Cramér-Rao bound. Similarly, in case of quantum multi-parameter estimation problems, there exist various figures of merit for example, the multi-parameter quantum Cramér-Rao bound, the Holevo Cramér-Rao bound, etc. All of these lower bounds are independent of the measurements performed on the quantum state as well as the estimators that are used in the problem. Thus, if one is able to find the optimal probe state which leads to the minimum possible error in the estimation, over the entire space of states, then the ultimate precision with which the parameter(s) of interest can be estimated, is completely determined. In quantum single-parameter estimation problems, the optimal probe is always the one which leads to the maximum possible value of the quantum Fisher information. However, performing a similar optimization in multi-parameter estimation problems is not as easy since the optimality of a particular probe may be dependent on how the covariances of the various estimators corresponding to the parameters of interest contribute to the total error in estimation. In this thesis, we study about the nature of optimal probe states which can be used to estimate two parameters encoded via SU(2) unitary parametrization processes. Using the tools of quantum multi-parameter estimation theory we find the explicit form of the optimal states for two and three-level quantum systems. Employing the Holevo Cramér-Rao bound in the case of single qubit systems and the multi-parameter quantum Cramér-Rao bound for single qutrit systems as fundamental bounds for the precision in simultaneous estimation of the two parameters, we show that in both situations the best probe is always a pure-state that maximizes the determinant of the quantum Fisher information matrix and is independent of the choice of the weight matrix which is used to defined the corresponding figure of merit. We also show that for both qubits as well as qutrits, the optimal probe

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